Sergei Mikhailovich Voronin

Sergei Mikhailovich Voronin

Sergei Mikhailovich Voronin ( Russian Сергей Михайлович Воронин , English transcription Sergei Mikhailovich Voronin; born March 11, 1946 in Gorno-Altaisk ; † October 18, 1997 in Moscow ) was a Russian number theorist .

Life

Voronin's father was a petroleum engineer and his mother a teacher. He grew up in Buguruslan in the Orenburg area . He studied piano at a music school, successfully took part in the mathematics Olympiads as a pupil, attended mathematical summer schools in Moscow and in 1963 switched to a special boarding school for mathematics in Moscow. From 1964 he studied at Lomonossow University , where he specialized in analytical number theory with Anatoly Alexejewitsch Karazuba (Karatsuba). In 1972 he received his doctorate on the Riemann zeta function . 1977 followed the habilitation at the Steklow Institute (via the Dirichlet zeta function). He was a professor of number theory at the State Pedagogical Institute in Moscow.

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Woronin proved in his dissertation that the Riemann zeta function does not obey a continuous differential equation. In 1975 he proved his universality theorem (part of his habilitation thesis) that an (arbitrary) continuous, non-vanishing analytic function in a circular disk can be approximated by the Riemann zeta function within the critical strip . The theorem shows the chaotic behavior of the Riemann zeta function in the critical strip. He also dealt with the zero point distribution of other zeta functions (Dirichlet, Epstein ). For example, in 1980 he showed that certain functions (in this case the Davenport-Heilbronn function, soon afterwards for some Epstein zeta functions), which are defined in the right half-plane by a Dirichlet series and fulfill a functional equation like the Riemann zeta function, but for which the Riemann hypothesis does not apply, but still have an abnormal accumulation of zeros on the critical straight line. In addition to problems related to the Riemann Hypothesis, he also dealt with additive number theory and applications of number theory in numerical mathematics (multi-dimensional numerical integration and interpolation ). He was also interested in the history of mathematics. ${\displaystyle {\tfrac {1}{2}}<\mathrm {Re} \,s<1}$

Voronin's universality theorem

Be a continuous function , which in the circular disk with has no zeros and inside the circular disk analytically is. Then for each there is a positive real number such that ${\displaystyle f}$ ${\displaystyle D(r)}$ ${\displaystyle 0<r<{\tfrac {1}{4}}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \tau }$

${\displaystyle |\zeta (s+{\frac {3}{4}}+i\tau )-f(s)|<\epsilon }$

holds for and with the Riemann zeta function . ${\displaystyle |s|\leq r}$ ${\displaystyle \zeta (s)}$

The theorem also applies to general Dirichlet-L functions . The theorem can also be formulated in such a way that continuous, analytical functions that do not vanish in circular disks , where those lie in the strip , can be approximated with any precision by translating the Riemann zeta function along the imaginary axis . Bhaskar Bagchi, for example, generalized it from circular disks to areas that are simply connected and compact and lie in a strip . ${\displaystyle D}$ ${\displaystyle f}$ ${\displaystyle D}$ ${\displaystyle {\tfrac {1}{2}}<\mathrm {Re} \,s<1}$ ${\displaystyle \zeta (s+i\tau )}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle {\tfrac {1}{2}}<\mathrm {Re} \,s<1}$

The Riemann hypothesis is equivalent to the theorem that the Riemann zeta function itself can also be approximated uniformly in the sense of Woronin's universality theorem.

Fonts

• with AA Karatsuba The Riemann Zetafunction , De Gruyter 1992 ISBN 978-3-11-013170-3

Web links

• Russian Biography, Russian Math. Surveys
• Voronin at Numbertheory Web, according to Russian Mathematical Surveys 1998

Individual evidence

1. ^ Voronin sentence on the universality of the Riemann zeta function , Izvestija Akad. Nauka, Volume 39, 1975, pp. 475-486 (Russian), English translation Math. USSR Izv., Volume 9, 1975, p. 443
2. ↑ Voronin's universality theorem at Mathworld
3. ^ Bagchi A joint universality theorem for Dirichlet L-Functions , Mathematische Zeitschrift, Volume 181, 1982, pp. 319-335
4. ^ Bagchi, "Recurrence in Topological Dynamics and the Riemann Hypothesis , Acta Math. Hungar., Vol. 50, 227-240, 1987